Invariant Feature Extraction Functions for UME-Based Point Cloud Detection and Registration

Point clouds are unordered sets of coordinates in 3D with no functional relation imposed on them. The Rigid Transformation Universal Manifold Embedding (RTUME) is a mapping of volumetric or surface measurements on a 3D object to matrices, such that when two observations on the same object are related by a rigid transformation, this relation is preserved between their corresponding RTUME matrices, thus providing linear and robust solution to the registration and detection problems. To make the RTUME framework of 3D object detection and registration applicable for processing point cloud observations, there is a need to define a function that assigns each point in the cloud with a value (feature vector), invariant to the action of the transformation group. Since existing feature extraction functions do not achieve the desired level of invariance to rigid transformations, to the variability of sampling patterns, and to model mismatches, we present a novel approach for designing dense feature extraction functions, compatible with the requirements of the RTUME framework. One possible implementation of the approach is to adapt existing feature extracting functions, whether learned or analytic, designed for the estimation of point correspondences, to the RTUME framework. The novel feature-extracting function design employs integration over $SO(3)$ to marginalize the pose dependency of extracted features, followed by projecting features between point clouds using nearest neighbor projection to overcome other sources of model mismatch. In addition, the non-linear functions that define the RTUME mapping are optimized using an MLP model, trained to minimize the RTUME registration errors. The overall RTUME registration performance is evaluated using standard registration benchmarks, and is shown to outperform existing SOTA methods.